WE PROVE THAT EVERY FINITE-EXPANSIVE HOMEOMORPHISM WITH THE SHADOWING PROPERTY HAS A KIND OF STABILITY. THIS STABILITY WILL BE GOOD ENOUGH TO IMPLY BOTH THE SHADOWING PROPERTY AND THE DENSENESS OF PERIODIC POINTS IN THE CHAIN RECURRENT SET. NEXT WE ANALYZE THE N-SHADOWING PROPERTY WHICH IS REALLY STRONGER THAN THE MULTISHADOWING PROPERTY IN CHERKASHIN AND KRYZHEVICH (TOPOL METHODS NONLINEAR ANAL 50(1): 125?150, 2017). WE SHOW THAT AN EQUICONTINUOUS HOMEOMORPHISM HAS THE N-SHADOWING PROPERTY FOR SOME POSITIVE INTEGER N IF AND ONLY IF IT HAS THE SHADOWING PROPERTY.